From Wikipedia, the free encyclopedia

A pinch is the compression of an electrically
conducting filament by magnetic forces. The conductor is
usually a plasma, but could also be a solid or
liquid metal. In a z-pinch, the current
is axial (in the z direction in a cylindrical coordinate
system) and the magnetic field azimuthal; in a
theta-pinch, the current is azimuthal (in the
theta direction in cylindrical coordinates) and the magnetic field
is axial. The phenomenon may also be referred to as a "Bennett
pinch"[1] (after
Willard Harrison Bennett),
"electromagnetic pinch",[2]
"magnetic pinch",[3] "pinch
effect"[4] or
"plasma pinch".[5]

History

The first creation of a z-pinch in the laboratory may have
occurred in 1790 in Holland when Martinus van Marum created an
explosion by discharging 100 Leyden jars into a wire.[26] The
phenomenon was not understood until 1905, when Pollock and
Barraclough[10]
investigated a compressed and distorted length of copper tube from
a lightning rod
after it had been struck by lightning. Their analysis showed that
the forces due to the interaction of the large current flow with
its own magnetic field could have caused the compression and
distortion.[27] A
similar, and apparently independent, theoretical analysis of the
pinch effect in liquid metals was published by Northrupp in
1907.[28]. The
next major development was the publication in 1934 of an analysis
of the radial pressure balance in a static z-pinch by Bennett[29] (See
the following section for details.)

Thereafter, the experimental and theoretical progress on pinches
was driven by fusion
power research. In their article on the "Wire-array z-pinch: a
powerful x-ray source for ICF", M G Haines et
al., wrote on the "Early history of z-pinches":[30]

In 1946 Thompson and Blackman [43] submitted a patent for a fusion reactor based on a toroidal z-pinch
[43][31] with
an additional vertical magnetic field. But in 1954 Kruskal and
Schwarzschild [44][32]
published their theory of MHD instabilities in a z-pinch. In 1956
Kurchatov gave his famous Harwell lecture showing nonthermal
neutrons and the presence of m = 0 and m = 1
instabilities in a deuterium pinch [45].[33] In
1957 Pease [46][34] and
Braginskii [47][35]
independently predicted radiative collapse in a z-pinch under
pressure balance when in hydrogen the current exceeds 1.4 MA. (The
viscous rather than resistive dissipation of magnetic energy
discussed above and in [32][36] would
however prevent radiative collapse). Lastly, at Imperial College in
1960, led by R Latham, the Plateau-Rayleigh instability was shown, and
its growth rate measured in a dynamic z-pinch [48].[37]"

Configurations

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One Dimensional
configurations

There are three analytic one dimensional configurations
generally studied in plasma physics. These are the θ-pinch, the Z-pinch, and the Screw Pinch.
All of the classic one dimensional pinches are cylindrically
shaped. Symmetry is assumed in the axial (z) direction and
in the azimuthal (θ) direction. It is traditional to name a
one-dimensional pinch after the direction in which the current
travels.

The θ-pinch

A sketch of the θ-Pinch Equilibrium. The z directed magnetic field
(shown in purple) corresponds to a θ directed plasma current (shown
in yellow).

The θ-pinch has a magnetic field traveling in the z direction.
Using Ampère's law (discarding the displacement
term)

Since B is only a function of r we can
simplify this to

So J points in the θ direction. θ-pinches tend to be
resistant to plasma instabilities. This is due in part to the
frozen in flux theorem, which is beyond the scope of this
article.

The Z-Pinch

A sketch of the z-Pinch Equilibrium. A -θ directed magnetic field
(shown in purple) corresponds to a z directed plasma current (shown
in yellow).

The Z-Pinch has a magnetic field in the θ direction. Again, by
electrostatic Ampere's Law

So J points in the z direction. Since
particles in a plasma basically follow magnetic field lines,
Z-pinches lead them around in circles. Therefore, they tend to have
excellent confinement properties.

The Screw Pinch The Screw pinch is an effort to
combine the stability aspects of the θ-pinch and the confinement
aspects of the Z-pinch. Referring once again to Ampere's Law

But this time, the B field has a θ component
and a z component

So this time J has a component in the z
direction and a component in the θ direction.

Two
Dimensional Equilibria

A toroidal coordinate system in common use in plasma
physics. The red arrow indicates the poloidal
direction (θ) and the blue arrow indicates the
toroidal direction (φ)

A common problem with one-dimensional equilibria based machines
is end losses. As mentioned above, most of the motion of particles
in a plasma is directed along the magnetic field. With the θ-pinch
and the screw-pinch, this leads particles to the end of the machine
very quickly (as the particles are typically moving quite fast).
Additionally, the Z-pinch has major stability problems. Though
particles can be reflected to some extent with magnetic
mirrors, even these allow many particles to pass. The most
common method of mitigating this effect is to bend the cylinder
around into a torus. Unfortunately this breaks θ symmetry, as paths
on the inner portion (inboard side) of the torus are shorter than
similar paths on the outer portion (outboard side). Thus, a new
theory is needed. This gives rise to the famous Grad-Shafranov equation.

The one dimensional equilibria provide the inspiration for some
of the toroidal configurations. An example of this is the ZETA
device at Culham England (which also operated as a Reversed
Field Pinch). The most well recognized of these devices is the
toroidal version of the screw pinch, the Tokamak.

Numerical solutions to the Grad-Shafranov equation have also
yielded some equilibria, most notably that of the Reversed
Field Pinch.

Three Dimensional
Equilibria

There does not exist a coherent analytical theory for
three-dimensional equilibria. The general approach to finding three
dimensional equilibria is to solve the vacuum ideal MHD equations.
Numerical solutions have yielded designs for stellarators. Some machines take advantage
of simplification techniques such as helical symmetry (for example
University of Wisconsin's Helically Symmetric eXperiment).

Formal
treatment

A stream of water pinching into droplets has been
suggested as an analogy to the electromagnetic pinch.[38] The
gravity accelerates free-falling water which causes the water
column to constrict. Then surface tension breaks the narrowing
water column into droplets (not shown here) (see Plateau-Rayleigh instability), which is
analogous to the magnetic field which has been suggested
as the cause of pinching in bead lightning.[39] The
morphology (shape) is similar to the so-called sausage instability in
plasma.

The Bennett
Relation

Consider a cylindrical column of fully ionized quasineutral
plasma, with an axial electric field, producing an axial current
density, j, and associated azimuthal magnetic
field, B. As the current flows through its own
magnetic field, a pinch is generated with an inward radial force
density of j x B. In a steady state with forces
balancing:

∇p = ∇(pe + pi) =
j x Β

where ∇p is the magnetic pressure gradient,
pe and pi is the electron and ion pressures.
Then using Maxwell's equation ∇ x
B = μ0j and the ideal gas lawp
= N k T, we derive:

(The Bennett Relation)

where N is the number of electrons per unit length
along the axis, Te and Ti
are the electron and ion temperatures, I is the total beam
current, and k is the Boltzmann constant.

The Generalized Bennett
Relation

The Generalized Bennett Relation considers a
current-carrying magnetic-field-aligned cylindrical plasma pinch
undergoing rotation at angular frequency ω. Along the axis of the
plasma cylinder flows a current density jz, resulting in
a toroidal magnetίc field Βφ. Originally derived by
Witalis,[40] the
Generalized Bennett Relation results in:[41]

where a current-carrying, magnetic-field-aligned cylindrical
plasma has a radius a,

J0 is the total moment of inertia with
respect to the z axis,

W⊥kin is the kinetic energy per unit length due to
beam motion transverse to the beam axis

WBz is the self-consistent Bz energy per
unit length

WEz is the self-consistent Ez energy per
unit length

Wk is thermokinetic energy per unit length

I(a) is the axial current inside the radius a
(r in diagram)

N(a) is the total number of particles per unit length

Er is the radial electric field

Eφ is the rotational electric field

The positive terms in the equation are expansional forces while
the negative terms represent beam compressional forces.

The
Carlqvist Relation

The Carlqvist Relation, published by Per Carlqvist in 1988,[42] is a
specialization of the Generalized Bennett Relation (above), for the
case that the kinetic pressure is much smaller at the border of the
pinch than in the inner parts. It takes the form

and is applicable to many space plasmas.

The Bennett pinch showing the total current (I) versus the number
of particles per unit length (N). The chart illustrates four
physically distinct regions. The plasma temperature is 20 K, the
mean particle mass 3×10-27 kg, and ΔWBz is
the excess magnetic energy per unit length due to the axial
magnetic field Bz. The plasma is assumed to be
non-rotational, and the kinetic pressure at the edges is much
smaller than inside.

The Carlqvist Relation can be illustrated (see right), showing
the total current (I) versus the number of particles per
unit length (N) in a Bennett pinch. The chart illustrates four
physically distinct regions. The plasma temperature is quite cold
(Ti = Te =
Tn = 20 K), containing mainly hydrogen with a
mean particle mass 3×10-27 kg. The thermokinetic energy
Wk >> π a2 pk(a).
The curves, ΔWBz show different amounts of excess
magnetic energy per unit length due to the axial magnetic field
Bz. The plasma is assumed to be non-rotational, and the
kinetic pressure at the edges is much smaller than inside.

Chart regions: (a) In the top-left region, the
pinching force dominates. (b) Towards the bottom, outward kinetic
pressures balance inwards magnetic pressure, and the total pressure
is constant. (c) To the right of the vertical line
ΔWBz=0, the magnetic pressures balances the
gravitational pressure, and the pinching force is negligible. (d)
To the left of the sloping curve ΔWBz=0, the
gravitational force is negligible. Note that the chart shows a
special case of the Carlqvist relation, and if it is replaced by
the more general Bennett relation, then the designated regions of
the chart are not valid.

Carlqvist further notes that by using the relations above, and a
derivative, it is possible to describe the Bennett pinch, the Jean's
criterion (for gravitational instability,[43] in
one and two dimensions), force-free magnetic fields,
gravitationally balanced magnetic pressures, and continuous
transitions between these states.

Crushing cans with the
pinch effect

Pinched aluminium can, produced from a pulsed magnetic field
created by rapidly discharging 2 kilojoules from a high voltage capacitor bank into a
3-turn coil of heavy gauge wire. Source: Bert Hickman, Stoneridge Engineering.

Many high-voltage electronics enthusiasts make their own devices
using pulsed power
techniques to produce a theta pinch capable of crushing an
aluminium soft drink can by pressure of strong magnetic field.

An electromagnetic aluminium can crusher consists of four main
components (1) A high
voltageDCpower supply which
provides a source of electrical energy
(2) A large energy dischargecapacitor to accumulate the electrical energy
(3) A high voltage switch or spark gap and (4) A robust coil (capable of surviving high magnetic pressure)
through which the stored electrical energy can be quickly
discharged in order to generate a correspondingly strong pinching
magnetic field (see diagram below).

Electromagetic pinch "can crusher": schematic diagram

In practice, such a device is somewhat more sophisticated than
the schematic diagram suggests, including electrical components
that control the current in order to maximize the resulting pinch,
and to ensure that the device works safely. For more details, see
the notes.[44]

Sam Barros's can crusher cost about $500, and uses a large SCR
and a 900 Volt capacitor bank
storing about 3000 Joules of
energy. For a very short time, it generates a magnetic field B~5T
(250,000 times the strength of the Earth's magnetic field) which
has magnetic pressure P ~ 100 atm. Rate of energy conversion (from
electric into magnetic and back) in this device is about 22 megawatts.[45]

Depictions

A fictionalized pinch-generating device was used in Ocean's Eleven, where
it was used to disrupt Las Vegas's power grid just long enough for
the characters to begin their heist.[46]

^R. S. Pease, "The
Electromagnetic Pinch: From Pollock to the Joint
European Torus", "Pollock Memorial Lecture for
1984 delivered at the University of Sydney, 28 November, 1984":
This review of the electromagnetic pinch starts with an exhibit
taken from Pollock's work, carefully preserved and drawn to
attention of modern research by Professor C. Watson-Munro. It is a
compressed and distorted length of copper tube originally part of
the lightning conductor on the Hartley Vale kerosene refinery in
New South Wales. It was known to have been struck by lightning.
Pollock and Barraclough (1905) from the Department of Mechanical
Engineering at Sydney University carried out an analysis to see
whether or not the compression could have arisen from the flow of
electric current. They concluded that the compressive forces, due
to the interaction of the large current flow with its own magnetic
field could have been responsible for the compression and
distortion. As far as I know, this is the first identified piece of
observational data on the electromagnetic pinch; and the first
theoretical discussion of the effect.

^
Northrupp E F 1907 "Some Newly Observed
Manifestations of Forces in the Interior of an Electric
Conductor" (1907) Phys. Rev. 24 474. He wrote: "Some
months ago, my friend, Carl Hering, described to me a surprising
and apparently new phenomenon which he had observed. He found, in
passing a relatively large alternating current through a
non-electrolytic, liquid conductor contained in a trough, that the
liquid contracted in cross-section and flowed up hill lengthwise of
the trough... Mr. Hering suggested the idea that this contraction
was probably due to the elastic action of the lines of magnetic
force which encircle the conductor... As the action of the forces
on the conductor is to squeeze or pinch it, he jocosely called it
the 'pinch phenomenon'.